Departamento de Matemáticas UAM

Seminario de Análisis y aplicaciones

Ponente: Anupam Gumber (University of Vienna)

Título: Fourier multipliers, function expansions and reproducing formulas associated to GTI systems

Resumen: 

For a given pair of frames {ψ_n} and {φ_n } in a separable Hilbert space H, the associated mixed frame operator S:H→H; f↦⟨f,ψ_n⟩φ_n is a bounded linear operator. In this talk, we discuss a characterization result for the mixed frame operator to be a Fourier multiplier which concerns a concept at the core of frame theory, namely, the reproducing formulas for frame pairs {ψ_n} and {φ_n} in H. The result turned out to be not only interesting in itself, but also important for further investigations. We discuss some properties of S and apply the obtained characterization to investigate reproducing (reconstruction) property when {ψn} and {φn} belong to generalized translation-invariant (GTI) systems, a special class of structured frame systems motivated by the utility of a recent notion considered in [2-4]. The concept of orthogonality plays a significant role in multiplexing techniques and in the synthesis of such frames, in this context, by utilizing the unitary extension principle of Christensen and Goh [1], we give a general construction for orthogonal GTI frame pairs in LCA group setting and derive explicit constructions for the B-splines generated frames.
References
[1] O. Christensen and S. S. Goh, The unitary extension principle for locally compact abelian groups with
co-compact subgroups. Proc. Amer. Math. Soc. 149(3) (2021), 1189-1202.
[2] J. W. Iverson, Subspaces of L2 (G) invariant under translation by an abelian subgroup. J. Funct. Anal.,
269(3) (2015), 865-913.
[3] M. S. Jakobsen and J. Lemvig, Reproducing formulas for generalized translation invariant systems on
locally compact abelian groups. Trans. Amer. Math. Soc., 368 (2016), 8447-8480.
[4] J. Lemvig and J. T. van Velthoven, Criteria for generalized translation-invariant frames. Studia Math.,
251(1) (2020), 31-63.

Hora y Lugar: Viernes 14 de Abril, 10:00-11:00, Aula 320, Módulo 17.